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Simple quantum mechanical system

In quantum mechanics, a two state system (also known as a two level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a system is two dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two state system can also be seen as a qubit. Two state systems are the simplest quantum systems that are of interest, since the dynamics of a one state system is trivial (as there are no other states the system can exist in). The mathematical framework required for the analysis of two state systems is that of linear differential equations and linear algebra of two dimensional spaces. As a result, the dynamics of a two state system can be solved analytically without any approximation. The generic behavior of the system is that the wavefunction's amplitude oscillates between the two states. A very well known example of a two state system is the spin of a spin 1/2 particle such as an electron, whose spin can have values +ħ/2 or −ħ/2, where ħ is the reduced Planck constant. The two state system cannot be used as a description of absorption or decay, because such processes require coupling to a continuum. Such processes would involve exponential decay of the amplitudes, but the solutions of the two state system are oscillatory.

Suppose that the system starts in one of the basis states at , say so that , and we are interested in the probability of occupation of each of the basis states as a function of time when is the time independent Hamiltonian. The probability of occupation of state i is In the case of the starting state, , and from above, Hence,

Obviously, due to the initial condition. The frequency is called the generalised Rabi frequency, is called the Rabi frequency, and is called the detuning. At zero detuning, , i. e., there is Rabi flopping from guaranteed occupation of state 1, to guaranteed occupation of state 2, and back to state 1, etc., with frequency As the detuning is increased away from zero, the frequency of the flopping increases (to Ω) and the amplitude of exciting the electron decreases to
For time dependent Hamiltonians induced by light waves, see the articles on Rabi cycle and rotating wave approximation.

Nuclear magnetic resonance (NMR) is an important example in the dynamics of two state systems because it involves the exact solution to a time dependent Hamiltonian. The NMR phenomenon is achieved by placing a nucleus in a strong, static field 'B'0 (the "holding field") and then applying a weak, transverse field 'B'1 that oscillates at some radiofrequency ωr. Explicitly, consider a spin 1/2 particle in a holding field and a transverse rf field 'B'1 rotating in the xy plane in a right handed fashion around 'B'0:

As in the free precession case, the Hamiltonian is , and the evolution of a state vector is found by solving the time dependent Schrödinger equation After some manipulation (given in the collapsed section below), it can be shown that the Schrödinger equation becomes

where and
As per the previous section, the solution to this equation has the Bloch vector precessing around with a frequency that is twice the magnitude of the vector. If is sufficiently strong, some proportion of the spins will be pointing directly down prior to the introduction of the rotating field. If the angular frequency of the rotating magnetic field is chosen such that , in the rotating frame the state vector will precess around with frequency , and will thus flip from down to up releasing energy in the form of detectable photons. This is the fundamental basis for NMR, and in practice is accomplished by scanning until the resonant frequency is found at which point the sample will emit light. Similar calculations are done in atomic physics, and in the case that the field is not rotating, but oscillating with a complex amplitude, use is made of the rotating wave approximation in deriving such results.

Two state systems are the simplest non trivial quantum systems that occur in nature, but the above mentioned methods of analysis are not just valid for simple two state systems. Any general multi state quantum system can be treated as a two state system as long as the observable one is interested in has two eigenvalues. For example, a spin 1/2 particle may in reality have additional translational or even rotational degrees of freedom, but those degrees of freedom are irrelevant to the preceding analysis. Mathematically, the neglected degrees of freedom correspond to the degeneracy of the spin eigenvalues. Another case where the effective two state formalism is valid is when the system under consideration has two levels that are effectively decoupled from the system. This is the case in the analysis of the spontaneous or stimulated emission of light by atoms and that of charge qubits. In this case it should be kept in mind that the perturbations (interactions with an external field) are in the right range and do not cause transitions to states other than the ones of interest.

Pedagogically, the two state formalism is among the simplest of mathematical techniques used for the analysis of quantum systems. It can be used to illustrate fundamental quantum mechanical phenomena such as the interference exhibited by particles of the polarization states of the photon, but also more complex phenomena such as neutrino oscillation or the neutral K meson oscillation. Two state formalism can be used to describe simple mixing of states, which leads to phenomena such as resonance stabilization and other level crossing related symmetries. Such phenomena have a wide variety of application in chemistry. Phenomena with tremendous industrial applications such as the maser and laser can be explained using the two state formalism. The two state formalism also forms the basis of quantum computing. Qubits, which are the building blocks of a quantum computer, are nothing but two state systems. Any quantum computational operation is a unitary operation that rotates the state vector on the Bloch sphere.